Tropical Cyclone Eye Thermodynamics

Home , Eye (cyclone), Inversion (meteorology)

VOLUME 126 MONTHLY WEATHER REVIEW DECEMBER 1998

H. E. WILLOUGHBY Hurricane Research Division, AOML/NOAA, Miami, Florida (Manuscript received 10 June 1997, in ®nal form 17 February 1998)

ABSTRACT In intense tropical cyclones, sea level pressures at the center are 50±100 hPa lower than outside the vortex, but only 10±30 hPa of the total pressure fall occurs inside the eye between the eyewall and the center. Warming by dry subsidence accounts for this fraction of the total hydrostatic pressure fall. Convection in the eyewall causes the warming by doing work on the eye to force the thermally indirect subsidence. Soundings inside hurricane eyes show warm and dry air aloft, separated by an inversion from cloudy air below. Dewpoint depressions at the inversion level, typically 850±500 hPa, are 10±30 K rather than the ϳ100 K that would occur if the air descended from tropopause level without dilution by the surrounding cloud. The observed temperature and dewpoint distribution above the inversion can, however, be derived by ϳ100 hPa of undilute dry subsidence from an initial sounding that is somewhat more stable than a moist adiabat. It is hypothesized that the air above the inversion has remained in the eye since it was enclosed when the eyewall formed and that it has subsided at most a few kilometers. The cause of the subsidence is the enclosed air's being drawn downward toward the inversion level as the air below it ¯ows outward into the eyewall. Shrinkage of the eye's volume is more than adequate to supply the volume lost as dry air is incorporated into the eyewall or converted to moist air by turbulent mixing across the eye boundary. The moist air below the inversion is in thermodynamic contact with the sea surface. Its moisture derives from evaporation of seawater inside the eye, frictional in¯ow of moist air under the eyewall, and from moist downdrafts induced as condensate mixes into the eye. The moist air's residence time in the eye is much shorter than that of the dry air above the inversion. The height of the inversion is determined by the balance between evaporation, in¯ow, and inward mixing on one hand and loss to the eyewall updrafts on the other.

1. Introduction tensi®es. Because the subsiding air is warmer than any other air in the cyclone, the descent must consume en- The lowest surface pressure in a hurricane coincides ergy released elsewhere in the storm. with the axis of vortex rotation inside the eye. The swirl- Balanced models predict, and observations con®rm, ing windÐthe tangential component of the wind vector that the most rapid pressure falls are con®ned to the in storm-centered cylindrical coordinatesÐincreases area inside the eyewall wind maximum (Shapiro and with distance outward from the axial stagnation point Willoughby 1982; Schubert and Hack 1982; Willoughby to the radius of maximum wind at the inner edge of the et al. 1982; Willoughby 1990). Tightening of the pres- eyewall. The eyewall, a ring of cumulonimbus convec- sure gradient across the wind maximum causes the wind tion, surrounds the eye and contains the sharpest radial to increase at, and inward from, the radius of maximum pressure gradient nearly coincident with strongest wind. wind so that the eyewall contracts as the wind strength- Because the eyewall slopes outward, the eye is approx- ens. Dynamically or thermodynamically forced out¯ow imately an inverted, truncated cone. The air aloft in the from the lower part of the eye into the eyewall causes eye is clear, warm, and dry, separated by an inversion the sinking and adiabatic warming, and hence the pres- from more moist, usually cloudy air near the surface sure falls. Inside the eye, where the wind increases with (Jordan 1952). Subsidence is the source of the eye's radius, air must subside from above to replace the loss warmth and dryness. Subsidence-induced adiabatic to the eyewall because horizontal motion is constrained warming increases the thickness between the ®xed tro- by the strong radial gradient of angular momentum. On popause height and the surface, lowering the hydrostatic the other hand, outside the eyewall, where the wind surface pressure at the vortex center as the cyclone in- decreases with radius and the radial angular momentum gradient is weaker, air can converge horizontally above the friction layer to replace the mass carried aloft by convection. This midlevel in¯ow from large radius sup- Corresponding author address: Dr. H. E. Willoughby, Hurricane Research Division AOML/NOAA, 4301 Rickenbacker Causeway, plies the angular momentum necessary to increase the Miami, FL 33143. swirling wind, in contrast with the frictional in¯ow near E-mail: [email protected] the surface, which extracts moist enthalpy from the sea

3053 3054 MONTHLY WEATHER REVIEW VOLUME 126 surface and feeds it into the convection, but can supply en subsidence. A conceptual model based upon this in- little excess angular momentum (Ooyama 1969, 1982). terpretation can synthesize observed eye soundings and The pressure fall between the eyewall and the center calculate realistic hydrostatic pressure falls from the accounts for only part of the total difference between eyewall inward to the axis of vortex rotation. the undisturbed surface pressure outside the storm and minimum sea level pressure (MSLP) at the center. The 2. Observed eye soundings process of bringing the late-summer tropical troposphere into thermodynamic equilibrium with the sea sur- Eastern Paci®c Hurricane Olivia formed from a dis- face at 28Њ±30ЊC, taking into account the elevation of turbance on the ITCZ on 19 September 1994. It reached equivalent potential temperature as the pressure de- hurricane intensity early on 24 September, less than two clines, can produce hydrostatic pressures as low as the days after it had become a tropical storm (Pasch and minimum sea level pressures of the most intense tropical May®eld 1996). Late on the 24th at 1922 UTC, the cyclones. Thermodynamic arguments (e.g., Miller 1958; National Oceanic and Atmospheric Administration's Emanuel 1986) that relate maximum possible intensity WP-3D research aircraft began two days of operations of hurricanes to sea surface temperature (SST) are es- in the hurricane, providing clear documentation of the sentially elaborations of this result. Nevertheless, the in¯uences of environmental wind shear and SST on in- moist adjustment process is only part of the story be- tensity and structure. When the aircraft arrived in Olivia, cause the tropospheric column in real tropical cyclones the MSLP had fallen to 949 hPa as the storm moved in is generally undersaturated except in organized con- weak easterly environmental wind shear over warm wa- vection, such as the eyewall, in the stratocumulus deck ter with SST Ͼ28ЊC. Olivia continued to intensify that caps the surface boundary layer, or in the out¯ow throughout the 4 h that the airplanes remained in the anvil near the tropopause. An airplane ¯ying outside of storm on the ®rst day. Subsequently, the MSLP appeared convection in the midtroposphere typically encounters to reach a minimum overnight at about 1200 UTC on precipitation, but little cloud. The eye itself is often the 25th. When the aircraft returned at 2021 UTC on clear, apart from boundary layer stratocumulus. This the 25th, Olivia's MSLP was 924 hPa, lower than on observation applies most consistently to intense tropical the previous day, but the storm ®lled throughout the rest cyclones that are continuing to intensify. Once inten- of the ¯ight in response to cooler SST and somewhat si®cation stops, but before the pressure has risen ap- stronger, then southwesterly, shear caused by an upper preciably, the eye typically ®lls with cloud (Jordan low northeast of the hurricane. By the time the aircraft 1961). left the storm on the second day, the MSLP had risen The conventional view of the eye's thermodynamics to 935 hPa. is that air detrains from the top of the eyewall and sinks A dropsonde observation in Olivia's eye at 2123 UTC inside the eye to the lower troposphere where it is en- on the 24th (Fig. 1a) is typical of intensifying tropical trained back into the eyewall. Inward mixing from the cyclones. It shows the expected inversion between 890 eyewall is hypothesized to force the subsidence and and 850 hPa, separating warm and dry air above from maintain the moisture and momentum budgets of the moist air below. In the moist air, the sounding follows subsiding air (Miller 1958; Malkus 1958; Holland a saturated adiabat down to 920 hPa and then a dry 1997). In this interpretation, the recirculation is rapid adiabat to the surface. Above the inversion, the dew- enough to replenish the eye's volume many times over point depression increases from 10 K at the top of the the hurricane's lifetime. The original argument for rapid inversion to 12 K at 600 hPa, the top of the sounding. replenishment of the air in the eye was that the calm The temperature and dewpoint soundings both run gen- air inside the eye did not appear to share in the trans- erally parallel to moist adiabats in the dry air, but with lation of the vortex as a whole (Malkus 1958), and that 1±3 K perturbations. Equivalent potential temperature, the eye moved by continuously reforming. More recent ␪e, has a weak minimum value of 350 K near 700 hPa observations from aircraft equipped with inertia navi- (Fig. 1b). It increases abruptly from 355 K at the top gation equipment show clearly that the low-level wind of the inversion layer to nearly 365 K at the bottom. is a superposition of circulation about the axis of ro- The vapor mixing ratio decreases from 11 gm kgϪ1 just tation and the translation of the axis (Willoughby and above the inversion to about 7 gm kgϪ1 at 600 hPa. The Chelmow 1982) so that it is kinematically possible for mixing ratio jumps to Ͼ20 gm kgϪ1 downward across a mass of air to move with the eye. the inversion. Here observed eye soundings are examined; they sug- Saturation point analysis (Betts 1982) is a good tool gest that the dry air above the inversion has a long for understanding eye thermodynamics. In undersatu- lifetime inside the eye, experiences only a few kilo- rated air, the saturation point is essentially the lifting meters total subsidence, and mixes only weakly with condensation level determined by expansion of the par- the moist air from the eyewall. The air below the in- cel along a dry adiabat, keeping constant water vapor version seems, by contrast, to derive largely from the mixing ratio, until saturation is reached at pressure PSAT eyewall, or from frictional in¯ow layer below it, through and temperature TSAT. In saturated air with suspended a complicated process of mixing and evaporatively driv- condensate, the saturation point is determined by com- DECEMBER 1998 WILLOUGHBY 3055

FIG. 1. (a) Skew T±logp diagram of the eye sounding in eastern Paci®c Hurricane Olivia at 2123 UTC on 24 September 1994. Isotherms slope upward to the right; dry adiabats slope upward to the left; moist adiabats are nearly vertical curving to the left. Solid and dashed curves denote temperature and dewpoint, respectively. The smaller dots denote saturation points computed for the dry air above the inversion, and the two larger dots temperature observed at the innermost saturated point as the aircraft passed through the eyewall. (b) Equivalent potential temperature, ␪e, water vapor mixing ratio, and saturation pressure difference, P Ϫ PSAT, as functions of pressure at 2123 UTC. (c) skew T±logp diagram of the eye sounding in Olivia at 2211 UTC on the 25th; and (d) ␪e, mixing ratio, and P Ϫ PSAT at 2211 UTC. 3056 MONTHLY WEATHER REVIEW VOLUME 126 pression of the parcel along a moist adiabat until all of the middle sounding. Although the air below the in- the suspended moisture has evaporated. The saturation version is saturated, P Ϫ PSAT is plotted here (and point de®nes an air parcel's thermodynamics uniquely subsequently) as though it were zero because no mea- and is conserved under adiabatic processes. The satu- surements of liquid water content are available. The ration points of mixtures of parcels lie along a ``mixing dewpoint depression at the top of the sounding is 3 line'' that joins the saturation points of the original par- K greater than on the 24th. The temperature and dew- cels on a thermodynamic diagram. point soundings form a ``Y'' shape that characterizes Saturation points for parcels above the inversion in all three soundings on the 25th, in contrast with the Olivia are indicated by dots in Fig. 1a. They lie within earlier sharp inversion. The soundings show a clear a degree or two of the moist adiabat that would pass progression: as the MSLP rose from 930 to 937 hPa through 25ЊC at 1000 hPa, corresponding to ␪e ϭ 350 in 2.7 h, the base of the inversion rose from 830 to K. The saturation pressure difference, P Ϫ PSAT in Fig. 740 hPa with about half of the ®lling, but most of the 1b [de®ned using the sign convention of Betts and Silva inversion's ascent, occurring in the hour between the Dias (1979), the negative of that used by Betts (1982)], ®rst two soundings. In all three soundings, the max- is 125 hPa above the inversion and decreases slowly imum P Ϫ PSAT was 150 hPa and occurred near 600 with altitude. In the moist air below the inversion, P Ϫ hPa. This value is 25 hPa greater than the maximum PSAT is small, but not zero or negative, because the sonde on the 24th and 250 hPa higher in the tropospheric apparently fell through a hole in the low-level strato- column than the level (850 hPa) where the greatest P cumulus. Ϫ P occurred the day before. The value of P Ϫ Flight level measurements made as the aircraft SAT PSAT at 600 hPa is also 40 hPa greater than the value passed through the eyewall provide estimates of the of P Ϫ P at the same altitude on the 24th. The saturation points there. These estimates, indicated by SAT greater P Ϫ PSAT near the top of the sounding re¯ects larger dots in Fig. 1, are the temperature and pressure continuing descent in the eye as Olivia intensi®ed of the last saturated observations on entry to the eye overnight. The temporally constant maximum P Ϫ or the ®rst saturated observation on exit during the PSAT on the 25th suggests that low-level moistening traverse of the eye when the sonde was deployed. They of the eye was caused largely by inward mixing rather should lie close to a moist adiabat that passes through than by ascent due to convergence below the inver- cloud base and should be close to T and P for SAT SAT sion. Comparison of the loci of saturation points be- eyewall air because most of the suspended condensate tween the two days is consistent with this interpre- migrates outward across the outward sloping eyewall tation. On the 24th the saturation points lay to the left updraft to its outer edge where it falls out. The ¯ight of the ␪ ϭ 350 K moist adiabat; on the 25th they level eyewall ␪ is typically ϳ10 K warmer than ␪ in e e e had migrated to the right of the same moist adiabat the eye, and the vapor mixing ratio is 10 gm kgϪ1 more. These measurements probably underestimate the dif- and approached the eyewall saturation points. The sat- ferences between the eye and eyewall because of sensor uration points in the lower troposphere, where mixing wetting and dilution of the updraft near the boundary. was strongest, migrated farthest to the right. The cool- For undilute saturated parcels with some suspended ing below 600 hPa can account for only about 1 hPa condensate, the saturation points are determined by of the 7 hPa total pressure rise so that some other moving down the eyewall moist adiabatÐwhich nearly processÐcooling above 600 hPaÐmust have been re- parallels the actual eyewall soundingÐuntil the sus- sponsible for the higher hydrostatic central surface pended liquid has evaporated. About 20 hPa of com- pressure. pression are required to evaporate a gram of condensate Another possible mechanism for moistening the eye per kilogram of dry air at the temperatures and pressures is evaporation of virga that falls into the eye, as de- typical of the midtroposphere in hurricanes. Saturation scribed in Hurricane Edna (Kessler 1958). This process points of mixtures between the cloudy eyewall air and would move the saturation points down a moist adiabat, dry air from the eye would lie along mixing lines that essentially parallel with the sounding that they de®ne. join the saturation points of the eye and eyewall. Thus, Since the process would be nearly adiabatic, ␪e would outward mixing of dry eye air would displace the sat- not change. The heat of evaporation would come pri- uration points in the cloudy eyewall updraft to the left, marily from the air into which the virga fell, not from away from the moist adiabat characterizing the eyewall, the hydrometeors. Thus, adiabatic evaporation would toward the locus of saturation points inside the eye. not leave a clear signature in the saturation points, and Similarly, inward mixing of cloudy eyewall air would P Ϫ PSAT would underestimate the total descent. Nev- move the saturation points in the eye to the right, toward ertheless, low re¯ectivity inside the eye and sparse vi- the eyewall sounding. sual or photographic observations of virga at radii well On the next day, the 25th, the soundings showed a inward from the eyewall argue generally against evap- clear signature of moisture mixing into the eye as the oration of virga as a major contributor to the eye's mois- storm ®lled. Three soundings were taken that day at ture budget. Snow or graupel falling into an eye covered 2106, 2211, and 2349 UTC. Figures 1c and 1d show with anvil cloud would be more dif®cult to observe DECEMBER 1998 WILLOUGHBY 3057

FIG. 2. As in Fig. 1 but for eastern Paci®c Hurricane Jimena: (a) skew T±logp; and (b) ␪e, mixing ratio, and P Ϫ PSAT, at 2058 UTC on

23 September 1991. (c) skew T±logp; and (d) ␪e mixing ratio, and P Ϫ PSAT at 2330 UTC the same day. visually or by radar so that this mechanism cannot be 1992). The P-3s ¯ew in Jimena on the 23d and 24th. discounted completely. Jimena's eye sounding at 2058 UTC on 23 September Hurricane Jimena formed from an African wave that 1991, when Jimena's MSLP was 945 hPa (Fig. 2a), is traversed Central America into the eastern Paci®c, de- almost the same as the ®rst sounding in Olivia. It shows veloped into a tropical depression on 20 September a sharp inversion near 850 hPa with warm, dry air aloft 1991, became a tropical storm on the 21st, and reached and moist air below. As before, the sounding in the moist hurricane strength on the 22d (Rappaport and May®eld air follows a moist adiabat downward from the inversion 3058 MONTHLY WEATHER REVIEW VOLUME 126 and then a dry adiabat to the surface; the temperature when mixing and in¯ow predominated. The time be- and dewpoint soundings in the dry air run roughly par- tween the soundings is much longer than the observed allel moist adiabats; and the dewpoint depressions in period of the convection so that they are random samples the dry air are about 10 K. The locus of the saturation of different phases of the oscillation, if this interpre- points above the inversion lie more or less along the tation is correct. same ␪e ϭϳ350 K moist adiabat as in Olivia. Equiv- Hurricane Hugo formed in the Atlantic from an Af- alent potential temperature, ␪e, has a minimum just rican wave. It reached hurricane intensity on 13 Sep- above the inversion (Fig. 2b) with a slow increase with tember 1989 east of the Leeward Islands and intensi®ed height above the inversion and then an abrupt increase rapidly, becoming a category 5 hurricane on the 15th from 350 K to nearly 370 K downward through the (Case and May®eld 1990). The eye sounding in Hugo inversion. The minimum ␪e is Ͼ10 K cooler than ␪e in (Fig. 3a) at 1900 UTC on 15 September 1989, when the eyewall, which is, in turn, about ϳ5 K cooler than Hugo's MSLP was 922 hPa, is qualitatively much the the warmest value below the inversion. The saturation same as Olivia's and Jimena's. In this case, the inversion pressure difference is 130 hPa just above the inversion is higher and thicker, extending from 790 hPa to 700 and decreases slowly with altitude. hPa. As in Jimena, the lowest ␪e, 350 K, and largest Some of the depression of ␪ e in the eye may result value of P Ϫ PSAT, 175 hPa, lie at the top of the inversion from radiative cooling of a degree or two per day, (Fig. 3b); and the locus of the saturation points straddles which can move the saturation points down a dry the ␪e ϭ 350 K moist adiabat. Dewpoint depressions in adiabat 100±200 m a day, or several hundred meters the dry air inside the eye are nearly 20 K. The tem- to a kilometer over the lifetime of a typical eye. This perature and dewpoint pro®les are only a little more sounding also shows evidence of some mixing from stable than moist adiabats. Inward mixing from the eye- the eyewall. The narrowing of the dewpoint depres- wall between 550 and 630 hPa, appears to have caused sion at 620±650 hPa clearly stems from moistening a5Њ decrease in dewpoint depression, saturation points' and cooling as condensate mixes into the eye and displacement to the right of the ␪e ϭϳ350 K moist evaporates. The saturation points for these levels are adiabat, and a 2±3 K increase in ␪e. displaced to the right toward their counterparts in the In the eyewall at 500 hPa, ␪e was 362 K based upon eyewall, and ␪ e increases to 355 K, still ϳ7 K cooler ¯ight-level temperature of 2ЊC. At 890 hPa the eyewall, than the eyewall. Just above this level, the tempera- ␪e was much warmer, 375 K measured as the aircraft ture increases and air dries, indicating subsidence, ¯ew through the base of a 20 m sϪ1 updraft in severe perhaps related to a 7 m sϪ1 downdraft that the aircraft turbulence. The difference between these values is con- data show in the south eyewall at 2258 UTC. sistent with the innermost saturated thermodynamic

On the 23d and 24th, Jimena maintained nearly con- measurement's being an underestimate of the updraft ␪e, stant intensity in southwesterly shear over warm water. although entrainment of low ␪e air into the updraft be- The only other sounding obtained in Jimena was 2.5 h tween 890 and 500 hPa may have lowered ␪e at the after Fig. 2a at 2330 UTC (Fig. 2c). As the MSLP rose higher altitude. to 949 hPa, the base of the inversion remained at the Super Typhoon Flo formed in the monsoon trough, same level, but the inversion layer became thicker. It became a typhoon on 15 September 1990, and inten- extended from 860 hPa to 810 hPa so that the sounding si®ed rapidly to super typhoon the next day (Joint Ty- became ``Y'' shaped somewhat like Olivia's eye sound- phoon Warning Center 1990). Figure 4a shows the eye ings on 25 September 1994. The saturation points in air sounding in Flo at 0500 UTC on 17 September 1991, formerly at the bottom of the dry layer moved to the near the end of the rapid intensi®cation, when Flo's right as the thickening inversion encroached on their MSLP was 891 hPa. The sounding extends through a level, and ␪e at 850 hPa increased from 350 to 365 K, greater depth of the atmosphere, from the surface to 275 the same value observed at 700 hPa in the eyewall. The hPa. The inversion lies from 720 to 760 hPa. The sound- saturation points above the inversion moved rightward ing below the inversion parallels a moist adiabat, but as, despite the simultaneous cooling, inward mixing the air is not particularly moist. Its dewpoint depression moistened the air enough to wipe out the ␪e minimum is 10ЊC. In the lowest 50 hPa above the surface, the (Fig. 2d). sounding is dry adiabatic. The relatively moist layer Since the structure and intensity of the storm re- may be a ``fossil'' inversion that was once much like mained fairly steady throughout both days, these sound- those shown in Figs. 1±3 but was subsequently subjected ings may represent different phases of an oscillating to an additional 100 hPa of subsidence. system rather than part of a long-term change. Eyewall Above the inversion, the dewpoint depression in- convection in Jimena was cyclic with a period of one- creases to 15ЊC. This part of the sounding also parallels half hour, close to the orbital period for air moving a moist adiabat up to 425 hPa. Above 425 hPa, ␪e is around the eye at the radius of maximum wind. The nearly constant at 370 K. This air appears to have been earlier sounding may represent a time when a recent drawn or mixed into the eye in the upper troposphere. pulse of convection had forced subsidence, whereas the Its thermodynamic properties are consistent with origin later sounding may represent an interval between pulses in the eyewall followed by descent that reaches only to DECEMBER 1998 WILLOUGHBY 3059

FIG. 3. As in Fig. 1 but for Atlantic Hurricane Hugo at 1839 UTC on 15 September 1989: (a) skew T±logp; and (b) ␪e mixing ratio, and

P Ϫ PSAT. a bit below 400 hPa, not to the surface as in the con- K moist adiabat increases from about 1±2 K to more ventional model. The saturation points de®ne a sounding than 10 K. The minimum ␪e in the sounding, 353 K, slightly warmer than the ␪e ϭϳ350 K moist adiabat. occurs in two places: at the top of the inversion and Above 600 hPa, the difference between TSAT and the 350 also at 840 hPa, the top of the near-surface dry adiabatic

FIG. 4. As in Fig. 1 but for Super Typhoon Flo at 0452 UTC on 17 September 1990: (a) skew T±logp with no eyewall temperature

indicated; and (b) ␪e, mixing ratio, and P Ϫ PSAT. 3060 MONTHLY WEATHER REVIEW VOLUME 126

TABLE 1. Kinematic calculation of rave, the vertically averaged sep- ematic center. Given that all of the eyes in question were aration between the falling drop sondes in Figs. 1±4, and the axis of Ͼ10 km in radius, it seems likely that these sondes fell vortex rotation, based upon ␷ave ϭ ␲rave⌬␭/(180Њ⌬t), where ␷ave is the average wind speed reported by the sonde in time interval ⌬t, in the region of ¯at horizontal gradients near the center. during which the wind direction changed by amount ⌬␭. Although the sonde in Olivia at 2123 UTC on the 24th failed to report the winds necessary to calculate r , the Storm Date/time ␷ (m sϪ1) ⌬␭(Њ) ⌬t (s) r (m) ave ave ave sharp inversion and low humidity in Figs. 1a and 1b Olivia 24/2123 No winds are consistent with descent near the axis. Finally, radar 25/2211 8 120 444 1698 observations of Olivia, Jimena, and Hugo showed pul- Jimena 23/2058 7 90 667 2971 23/2330 2 420 389 106 sating convection on a timescale comparable with the Hugo 15/1839 23 255 822 4249 orbital period of air moving with the maximum swirling Flo 17/0452 6 110 867 2709 wind. Since this period is not too different from the Brunt±VaÈisaÈlaÈ period, it would not be surprising if the convection induced axisymmetric, gravity wave oscil- layer. The largest saturation pressure difference, 150 lations, perhaps with amplitudes as large as tens of me- hPa, coincides with the upper ␪e minimum at 720 hPa. ters. Apart from the differences between the two sound- Unlike Figs. 1±3, there is no clear evidence of moisture ings in Jimena, the data show no obvious signatures of being mixed into Flo's eye in the lower troposphere. this kind of noise, but it is certainly possible that aliased These soundings are unusual only in that they were vertical oscillations affect the thermal structure ob- selected for well-de®ned inversions somewhat below served in eye soundings. ¯ight level. Often reconnaissance aircraft operating at 700 hPa ¯y either at the inversion or in the moist layer, 3. Conceptual model even in intense tropical cyclones. For example, throughout the rapid intensi®cation of Hurricane Opal of 1995, How does the two-layer thermodynamic structure of the eye soundings below ¯ight level at 700 hPa were the eye come about? Measurements of chemical tracers essentially moist adiabatic. Only after Opal attained suggest long residence times for air in the upper reaches minimum pressure, 916 hPa, did the inversion descend of the eye (Newell et al. 1996). Figure 5 illustrates a below ¯ight level. Then, as the storm ®lled, the inver- hypothetical model of the eye in which the air above sion rose and thickened. The correlations between low- the inversion has remained in the eye since it was en- ering of the inversion, warming and drying of the eye, closed when the eyewall formed. Under this hypothesis, and intensi®cation on one hand and moistening of the streamlines emerge from the eyewall near the tropo- eye and ®lling on the other seem to be common, but pause, bend downward inside the eye, and rejoin the not inevitable. eyewall in the lower troposphere; but the motion is so Several considerations affect interpretation of eye- slow that the air parcel trajectories do not close. When sounding data: representativeness of individual sound- the eyewall ®rst formed, it enclosed air from the cloud ings, location of the sounding within the eye, and pos- mass of the preexistent tropical storm. The sounding sible short-term (period ϳ1 h) oscillations of the eye as inside might plausibly have been saturated, or nearly a whole. Although no examples of multiple simulta- so, and dominated by convective adjustment toward a neous dropsonde deployments in the eyes of tropical moist adiabat near ␪e ϭϳ350 K. Thus, the observed cyclones seem to exist, horizontal pro®les of ¯ight-level P Ϫ PSAT may represent the total subsidence since the temperature and dewpoint usually show relatively ¯at eye formed. In this interpretation, air inside the eye sinks gradients within about half an eye radius of the center gradually as loss to the eyewall draws mass outward and steeper gradients close to the eyewall. It would be around the bottom of the eye. Dewpoint depressions at worthwhile to drop sondes in pairs to assess repeat- the inversion are 10±30 K rather than the ϳ100 K that ability of thermodynamic measurements and also as would occur if the air originated at the tropopause and tracers for the kinematics. Still, if it can be established descended without dilution. The local minimum of ␪e that the sonde fell near the eye center, the measurements is consistently in the lower troposphere. Its value, Ͼ10 should be representative. Sondes usually spiral down- K below the observed eyewall ␪e, is dif®cult to explain ward as they fall through air circulating around the vor- if one supposes that the other properties of the air derive tex axis. The wind speed and changes of wind direction from mixing with eyewall air. The hypothesis of 100± reported by the sondes can be related kinematically to 200-hPa dry adiabatic forced descent inside the eye the distance from the axis of rotation eliminates the need for hypothetically large inward mixing of eyewall air to force subsidence and maintain the ␷ ϭ ␲r (⌬␭/180Њ⌬t), ave ave moisture budget inside the eye (Malkus 1958; Miller where rave is the average distance from the axis and ␷ ave 1958). is the average wind speed reported during time interval An average sounding for the environment of Atlantic

⌬t as the wind direction changed by ⌬␭ degrees. Table Hurricanes (Sheets 1969) typically shows minimum ␪e 1 shows results of this calculation that con®rm that all of 339 K at 3±6-km altitude. When the soundings are sondes with reported winds fell within 5 km of the kin- strati®ed by surface pressure, the mean of soundings DECEMBER 1998 WILLOUGHBY 3061

FIG. 5. Schematic illustration of the secondary ¯ow in the eye and eyewall of a hurricane. The frictional indraft feeds the buoyancy-driven primary updraft and out¯ow in the eyewall cloud. In¯ow under the eyewall is derived from convective downdrafts. It, and evaporatively driven descent along the inner edge of the eyewall, feed moist air into the volume below the inversion. Gradual thermodynamically or dynamically driven descent of dry air inside the eye warms the air column adiabatically. The descent is forced as convection draws mass from the bottom of the eye into the eyewall. Balance between moist-air production and loss to the eyewall determines the rate of rise or fall of the inversion. with surface pressures Ͼ1010 hPa shows lower mini- or no detrainment from the eyewall into the eye in the mum ␪e, 335 K at 3-km altitude; the mean of soundings upper troposphere. The convergence due to shrinking t, where A ϭ ␲a 2 isץ/aץt ϭ 2aϪ1ץ/Aץwith surface pressures Ͻ995 hPaÐwhich are more of the eye is AϪ1 strongly in¯uenced by convectionÐhas minimum ␪e ϭ the area for an eye of radius a at some speci®ed altitude t. If during some time interval, theץ/aץt ϭ 2␲aץ/Aץ K, also at 3 km. In the latter soundings, ␪e a ki- and 342 lometer or two above the minimum is 345±346 K, still area of the eye were to contract by a half or a third over about 5 K cooler than the minimum ␪e observed in the the depth between 850 and 150 hPa and all of the excess eyes described in section 2. If the Atlantic mean en- air were to be squeezed downward through the 850-hPa vironmental soundings with surface pressures Ͻ995 hPa surface, the subsidence at the bottom of the volume may be taken as representative of the air enclosed during would be 500±700 hPa, 3±5 times the subsidence ex- formation of the eyes of these, predominantly Paci®c, pected from the saturation pressure de®cits in Figs. 1± tropical cyclones, 5±6 km of descent combined with 4. The question seems to be not whether eye contraction evaporation of about 10 gm kgϪ1 of virga would be can supply enough air to explain the subsidence, but required to bring 350 K air from 9 km (where this value rather how to dispose of the excess air supplied by even of ␪e is found) to the inversion level with the observed modest shrinking of the eye. temperature and mixing ratio. Alternatively, 40% di- The subsidence computed from the saturation pres- lution of 345 K eye air with 365 K eyewall air could sure differences is 100±150 hPa, equivalent to 1±1.5 produce the same result. Thus, based upon plausible km of descent. If all of the subsidence happened in two initial soundings, it is possible to argue for more descent days, the vertical velocity inside the eye would be ϳ1 and evaporation, or somewhat more mixing from the cm sϪ1. If the subsidence happened in 5 days, the ve- eyewall into the eye, or both. Nonetheless descent from locity would be 0.4 cm sϪ1. These ®gures are substan- tropopause height and rapid mixing into the eye appear tially smaller than the 11 cm sϪ1 subsidence determined to be inconsistent with observations. by Franklin et al. (1988) in Hurricane Gloria of 1985. As shown in Fig. 5, loss of mass to the eyewall might But the Gloria observations were unusual because they be balanced by shrinking of the eye's volume with little captured the downward plunge of the inversion as Glo- 3062 MONTHLY WEATHER REVIEW VOLUME 126 ria's MSLP fell 10 hPa in less than 5 h and because the analogous to that described in the semigeostrophic the- rapid descent was con®ned to a relatively shallow ver- ory of frontogenesis. Clearly, mixing into the eye must tical interval with comparable ascent above the sinking occur to prevent the collapseÐconsistent with obser- layer. Even though the interval between the soundings vations of the misty boundary layer and inclined cloud was 4 h, much longer than the typical period of variation rolls at the eyewall. If the collapse were not controlled, in eyewall convection, it is still possible that the large the swirling velocity would drop discontinuously to zero apparent subsidence re¯ected an aliased short-period os- just inside the radius of maximum wind, so that there cillation superimposed on long-term intensi®cation of would be no cyclostrophic height fall between the wind the hurricane. maximum and the center. Emanuel also offers a math- In seeming contradiction with the hypothesis of ematical argument that the isobaric height must fall most thermodynamic isolation from the eyewall, some mix- rapidly in the region of heating at the eye boundary as ing into the eye seems to occur. The shallow layers a ``proof'' that heating alone cannot raise the temper- of moistening and cooling, but increasing ␪ e , shown ature at the eye's center above that in the eyewall. in Figs. 2 and 3, are examples of one kind of mixing An example of the effect appears in a calculation of that is apparently too weak to dominate the moisture Hurricane Allen's intensi®cation [Fig. 16 of Willoughby budget. Another mechanism for moistening the eye et al. (1982)] with a frictionless Sawyer-Eliassen equa- involves small-scale mixing adjacent to the eyewall. tion that reproduces the observed wind increase and A mixture that contained clear air from the eye and geopotential decrease quantitatively [cf. Fig. 15 of Wil- moist air from the eyewall in proportions such that it loughby et al. (1982)] for heating rates consistent with was just saturatedÐmisty rather than cloudyÐwould the observed radar re¯ectivity. The computational re- be the coolest and densest possible for speci®ed prop- sults agree in detail with observed changes with two erties of the original air masses (Betts 1982). Jorgen- exceptions: In the calculations, the pressure falls 50% sen's (1984) analysis of in situ aircraft data revealed more rapidly just inside the eye than it does at the center, evaporatively driven descent along the inside edge of and the wind decreases with time around the center of the eyewall. Visual observation (e.g., Simpson 1954) the eye in response to the decreasing pressure gradient. and photographs of the eyewall from inside of the eye In the observations, the pressure falls most rapidly at (Fig. 6a) often show a descending cascade of misty the eye center, and the wind increases throughout the air. Evidently, evaporation of condensate mixed into eye, though only slowly at the center. the eye forms a moist descending boundary layer that An argument counter to Emanuel's recognizes that appears as a ¯ow of mist or pileus down the inside the pressure fall at the center is still a substantial fraction of the eye into the stratocumulus-topped moist layer. of that at the eyewall and that any point at the eyewall As an aircraft penetrates into the eye, it often expe- will soon be left outside the eye as a result of eyewall riences a ®nal shudder of turbulence when the eye is contraction. Thus, the instantaneous pressure falls are visible ahead through a veil of mist after the airplane more rapid at the eye boundary, but they do not last as has emerged from dense cloud. Visual and photo- long. The response to heating alone can force height graphic observations show the clouds on the inward falls throughout the eye as it makes the wind pro®le edge of the eye organized into inclined rolls that wind inside the eye increasingly ``u-shaped.'' Mixing is con- helically around the eye (Bluestein and Marks 1987) ®ned to a narrow layer of strong gradients next to the that may be a manifestation of shearing instability of eyewall where it prevents collapse of the gradients to a the radial wind pro®le (Emanuel 1984). Thus, the discontinuity and, incidently, mixes moisture into the misty layer is turbulent as well as moist. The fre- eye to drive the moist cascade at the eyewall. Although quently observed clear ``moat'' at low level just inside mixing is essential to maintenance of a continuous wind the eyewall (Fig. 6b, Simpson 1952; Simpson and pro®le required for thermally driven descent to occur Starrett 1955) may result from the moist descent's throughout the eye, it does not necessarily cause the ¯ow through the inversion into the moist air below. descent. Emanuel (1997) builds upon earlier work (Eliasen The moisture below the inversion thus seems to derive 1959; PalmeÂn and Newton 1969) to hypothesize that in part from condensate mixed across the inside edge heating in the eyewall causes a collapse toward a ther- of the eyewall, but the other sources may include sea± mal and momentum discontinuity at the eye boundary, air interaction inside the eye and frictional in¯ow under

→

FIG. 6. (a) Photograph inside the eye of Hurricane Olivia at 2136 UTC on 25 September 1993, showing the downward cascade of moist air along the inner edge of the eyewall. The moist boundary layer is most clearly visible along the border between the eyewall and the sky that runs from the left center to the upper right. The eyewall slopes away from the camera so that the line of sight is tangent to it along the border. This geometry renders the cascade more visible because it provides a long optical path in the moist air. The mist visible against the sky derives from cloud and virga mixed a kilometer or two into the eye. (b) Photograph of the hub cloud in Olivia's eye at 2137 UTC. The hub cloud and center of circulation occupy the lower right of the frame with the lower clouds that surround the hub extending from the lower left to the upper right and the clear moat in the upper-left corner (photographs by James Franklin, HRD/AOML). DECEMBER 1998 WILLOUGHBY 3063 3064 MONTHLY WEATHER REVIEW VOLUME 126 the eyewall. Jordan (1952) considered these sources, but saturation pressure difference at the top of the inver- rejected bothÐthe former on the basis of the supposed sion. The synthesis also requires a ``generating sound- lack of outward slope of the eyewall, and the latter ing,'' the sounding in the air that was enclosed, by because no one believed that vortex translation extended hypothesis, when the eye ®rst formed and that lies into the eye. It is true that frictional in¯ow toward the along the locus of the observed saturation points in eyewall in the boundary layer decelerates at the wind subsequent eye soundings. This sounding should gen- maximum and most of it turns upward. Nevertheless, erally be a bit more stable than the ␪ e ϭ 350 K moist the angular momentum budget of the boundary layer adiabat that passes through 25ЊC (298.16 K) at 1000 implies that as much as 25% of the in¯ow should pass hPa. It is generated by computation of a moist adiabat under the eyewall into the eye. If the eyewall works the corresponding to a 1000-hPa temperature of 298.16 same way as outer rainbands (Powell 1990), this in- K ϩ⌬T and then scaling that adiabat by the factor

¯owing air should be preferentially low ␪e downdraft 298.16/(298.16 ϩ⌬T ), where ⌬T is 3±5 K. Since air so that turbulent transfer from the surface must play moist adiabats' lapse rates decrease with increasing an essential role in its reaching the high values of ␪e ␪ e , the scaled adiabat will be more stable than the observed below the inversion. Similarly, if the only unmodi®ed moist adiabat passing through 298.16 K. source of moist air were converging moist descent along The eye sounding above the inversion derives from the inner edge of the eye above the surface, it would the generating sounding by dry adiabatic compression not be possible for it to converge toward very low an- equal to the observed P Ϫ PSAT at the inversion level gular momentum air at the eye's center without surface and decreasing linearly (or perhaps quadratically) to friction. As a result of heating from below and surface zero at the tropopause. Linear decrease is intuitively friction, the air below the inversion in the eye is prob- satisfying because it corresponds to uniform conver- ably turbulent enough for the moist air from the different gence inside the eye. Observed eye soundings below sources to be blended quickly. The balance between the inversion are moist adiabatic from the base of the production of moist air by mixing across the eye bound- inversion to the mixing condensation level and then ary, frictional in¯ow, and local turbulent exchange at nearly dry adiabatic to the surface. The synthetic the surface, on one hand, and loss of moist air to the soundings do not retain quite this much detail; they eyewall updrafts on the other, determines the depth of approximate the sounding below the inversion as a the inversion in the eye. If the inversion level rises or moist adiabat that extends from the temperature at the descends fairly slowly, the moist air must cycle quickly base of the inversion to the surface. It is also possible back into the eyewall to balance the fairly large sources, to specify that the sounding below the base of the so that its residence time in the eye is much shorter than inversion has a ®xed dewpoint depression as shown that of the dry air above the inversion. in Figs. 1 and 4. The convective updrafts in the eyewall are buoyant A synthetic sounding corresponding to Fig. 4 ap- with respect to the air around the eye, but not with pears in Fig. 7. For properly chosen values of the respect to the eye itself. Collectively, they act as a ``heat parameters, it is clearly possible to synthesize a given pump'' that does work on the eye by pulling air out sounding fairly accurately. Nevertheless, one should around the bottom of the eye to force thermally indirect be careful not to read too much into the result because descent. Since the updrafts do not rise in the warmest the synthesis is a quantitative description with a half- air inside the eye, the warm anomaly that causes the dozen disposable parameters rather than a predictive lowest hydrostatic pressure does not necessarily limit physical model. eyewall convection. When the convection is intense, net The synthetic eye soundings are useful for determi- ¯ux from the eye lowers the inversion and promotes nation of the hydrostatic pressure fall between the eye- descent above the inversion, warming and drying the wall and the axis of rotation consistent with the hypo- eye. When the convection weakens, net frictional in¯ow thetical eye thermodynamics. An upward hydrostatic in- and mixing into the eye raise the inversion, cooling the tegration along an assumed saturated adiabatic eyewall eye and ®lling it with cloud. This interpretation is con- sounding yields the 150-hPa height, and a downward sistent with the idea that convection causes the pressure hydrostatic integration along the synthetic eye sounding fall between the eyewall and the axis of vortex rotation can, if the eyewall surface pressure is chosen appro- through compensating subsidence and adiabatic warm- priately, recover the observed MSLP. The eyewall ing concentrated in the eye. sounding is the moist adiabat that corresponds to ␪e below the inversion so that the ®rst part of the calculation estimates the (dominant) contribution of convec- 4. Synthetic soundings tive warming to the total pressure de®cit. Iterative ad- The foregoing conceptual model provides a frame- justment of the eyewall surface pressure, initially as- work for generation of synthetic eye soundings based sumed to be the observed MSLP plus one-third of the upon temperature and dewpoint at the bottom of the difference between 1013 hPa and the MSLP, changes inversion, depth of the inversion, and the amount of the 150-hPa height and hence the result of the downward subsidence inside the eye, estimated as the maximum integration. This process converges reliably in a few DECEMBER 1998 WILLOUGHBY 3065

FIG. 7. Synthetic sounding in the eye of Super Typhoon Flo for the time and conditions shown in Fig. 4, illustrated (a) as a skew T±logp

diagram; and (b) as graphs of ␪e, mixing ratio, and P Ϫ PSAT. iterations to the observed MSLP because the central would expect that lowering of the inversion in nature pressure is nearly linear in the eyewall surface pressure. would occur with strong subsidence and greater cyclone Figure 8 illustrates the pressure decrease from the intensity, as is observed. But as far as the hydrostatic eyewall to the center of the eye for inversions at 900, calculation goes, it is primarily the maximum amount 700, and 500 hPa, subsidence at the top of the inversion of subsidence that determines the pressure fall from the from 0 to 200 hPa, and linear (n ϭ 1) or quadratic (n eyewall to the center of the eye. The maximum possible ϭ 2) decrease of subsidence with height above the top pressure fall for 150 hPa of descent is 20±25 hPa for a of the inversion. The pressure fall increases essentially linear decrease of subsidence with distance from the linearly with descent at the inversion top, and is not inversion and 30 hPa for a quadratic decrease. In the particularly sensitive to the level of the inversion. One former case, mass convergence is constant with altitude; in the latter case, it increases with altitude above the inversion so that more subsidence and more adiabatic warming occur through a greater depth. Pressure falls vary nearly linearly with subsidence for a given inversion height. The maximum pressure difference attainable through dry descent is consistent with the observation that a quarter to a third of the total pressure fall in intense tropical cyclones occurs inside the radius of maximum wind. Tropical cyclone intensi®cation thus appears as a two-stage process: Convec-

tion, with its roots in the high ␪e boundary layer, trans- ports moist enthalpy extracted from the sea upward to

elevate ␪e in the upper troposphere, producing a broad area of low pressure. As the wind increases, dry descent is increasingly localized inside the radius of maximum wind so that the pressure gradient at the edge of the eye tightens and the wind strengthens according to the well- FIG. 8. Pressure fall from the eyewall to the center of the eye as established convective ring model (Willoughby et al. a function of maximum subsidence for inversions at 900 hPa, 700 hPa, and 500 hPa, and for linear (n ϭ 1) or quadratic (n ϭ 2, 700 1982). The dry process does lower the pressure farther, hPa only) variation of subsidence with height above the inversion. but its most signi®cant effect is to sharpen the pressure 3066 MONTHLY WEATHER REVIEW VOLUME 126 gradients that sustain the strongest winds. This aspect near the bottom of the eye. It thus forces about a is illustrated clearly when a formerly intense hurricane centimeter per second of subsidence and gradual ad- loses its strong convection over cold water or as a result iabatic warming and drying of the eye. Warming of an eyewall succession. Well documented examples through this mechanism can account for no more than are Gloria of 1985 (Willoughby 1990) and Gilbert 30 hPa of pressure fall from the radius of maximum (Black and Willoughby 1992). Aircraft observations wind to the eye's center. showed little temperature anomaly at 700 hPa inside the eyes of these hurricanes after weakening. MSLP rose Acknowledgments. Discussions with P. Black, K. by 20 and 52 hPa, respectively, as the radial wind pro- Emanuel, and G. Holland informed my ideas on this ®les lost their sharp peak at the eyewall and became topic, though they may well disagree with some of my ¯at. The latter pressure rise is even more than one might interpretation. I also thank B. Albrecht, K. Emanuel, D. expect from reversal of the dry adiabatic warming inside Churchill, J. Franklin, J. Gamache, and G. Holland for the eye alone. thoughtful comments that greatly improved the manuscript. It is with sorrow that I note the recent passing of C. L. Jordan whose pioneering work has shaped my 5. Conclusions thought on eye thermodynamics since I ®rst read it as The soundings and interpretation presented here sug- a student. This research was supported under ONR gest strongly that the thermodynamics of tropical cy- Grant N00014-94-F-0045. clone eyes do not obey conventional models in which air detrains from the eyewall near the tropopause, sinks REFERENCES through most of the depth of the troposphere inside the eyeÐacquiring moisture and momentum as needed to Betts, A. K., 1982: Saturation point analysis of moist convective maintain its bulk propertiesÐand is entrained back into overturning. J. Atmos. Sci., 39, 1484±1505. the eyewall at the bottom. 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